is a Hilbert algebra in the sense of [1]}
Indeed, it is only matter of straightforward computations to check that the conditions which define the structure of a Hilbert algebra are satisfied.
Moreover, one can check that \(L_f\psi =f^-\ast \psi \)  for all \(f,\psi \in {L}^{1,2}_{B}\) , where \(f^-(x):=f(-x)\) . Then, after observing that the completion of \({L}^{1,2}_{B}\)  with respect to the norm induced by the scalar product \(\langle \,,\, \rangle _B\)  is exactly \(L^2\big (\mathbb {R}^2\big )\) , one
can prove that every element in
\({L}^2_B\)  is a left bounded operator associated to \({L}^{1,2}_{B}\)  according to [1]}. Moreover, in view of the effect of the involution \(J\) , every element in
\({L}^2_B\)  is also a right bounded operator [1]}.
Indeed, a direct computation shows

(JLfJ)(x) = R2 dy  f(y-x) B(y,x) (y)
= 12B2 R2 dy  -(y-x) B(x,y) f(y)
